Advertisements
Advertisements
Question
Prove that:
Advertisements
Solution
\[\text{LHS} = \frac{\left( 2n + 1 \right)!}{n!} \]
\[ = \frac{\left( 2n + 1 \right)\left( 2n \right)\left( 2n - 1 \right) . . . . \left( 4 \right)\left( 3 \right)\left( 2 \right)\left( 1 \right)}{n!}\]
\[ = \frac{\left[ \left( 1 \right)\left( 3 \right)\left( 5 \right) . . . . . . . . . \left( 2n - 1 \right)\left( 2n + 1 \right) \right]\left[ \left( 2 \right)\left( 4 \right)\left( 6 \right) . . . . . . . . . \left( 2n \right) \right]}{n!} \]
\[ = \frac{2^n \left[ \left( 1 \right)\left( 3 \right)\left( 5 \right) . . . . . . . . . \left( 2n - 1 \right)\left( 2n + 1 \right) \right]\left[ \left( 1 \right)\left( 2 \right)\left( 3 \right) . . . . . . . . . \left( n \right) \right]}{n!}\]
\[ = \frac{2^n \left[ \left( 1 \right)\left( 3 \right)\left( 5 \right) . . . . . . . . . \left( 2n - 1 \right)\left( 2n + 1 \right) \right]\left[ n! \right]}{n!}\]
\[ = 2^n \left[ \left( 1 \right)\left( 3 \right)\left( 5 \right) . . . . . . . . . \left( 2n - 1 \right)\left( 2n + 1 \right) \right] = \text{RHS}\]
\[ \text{Hence, proved} . \]
APPEARS IN
RELATED QUESTIONS
If P (n, 5) = 20. P(n, 3), find n ?
If P(11, r) = P (12, r − 1) find r.
In how many ways can five children stand in a queue?
From among the 36 teachers in a school, one principal and one vice-principal are to be appointed. In how many ways can this be done?
Four books, one each in Chemistry, Physics, Biology and Mathematics, are to be arranged in a shelf. In how many ways can this be done?
How many words, with or without meaning, can be formed by using the letters of the word 'TRIANGLE'?
There are 6 items in column A and 6 items in column B. A student is asked to match each item in column A with an item in column B. How many possible, correct or incorrect, answers are there to this question?
Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, if no digit is repeated? How many of these will be even?
How many different words can be formed with the letters of word 'SUNDAY'? How many of the words begin with N? How many begin with N and end in Y?
How many permutations can be formed by the letters of the word, 'VOWELS', when
there is no restriction on letters?
How many permutations can be formed by the letters of the word, 'VOWELS', when
each word begins with E?
How many permutations can be formed by the letters of the word, 'VOWELS', when
all vowels come together?
In how many ways can a lawn tennis mixed double be made up from seven married couples if no husband and wife play in the same set?
How many words (with or without dictionary meaning) can be made from the letters in the word MONDAY, assuming that no letter is repeated, if all letters are used but first is vowel.
Find the number of words formed by permuting all the letters of the following words:
INDEPENDENCE
Find the number of words formed by permuting all the letters of the following words:
ARRANGE
Find the number of words formed by permuting all the letters of the following words:
INDIA
Find the number of words formed by permuting all the letters of the following words:
SERIES
Find the total number of arrangements of the letters in the expression a3 b2 c4 when written at full length.
A biologist studying the genetic code is interested to know the number of possible arrangements of 12 molecules in a chain. The chain contains 4 different molecules represented by the initials A (for Adenine), C (for Cytosine), G (for Guanine) and T (for Thymine) and 3 molecules of each kind. How many different such arrangements are possible?
In how many ways can the letters of the word ASSASSINATION be arranged so that all the S's are together?
If the permutations of a, b, c, d, e taken all together be written down in alphabetical order as in dictionary and numbered, find the rank of the permutation debac ?
Find the total number of ways in which six ‘+’ and four ‘−’ signs can be arranged in a line such that no two ‘−’ signs occur together.
In how many ways can the letters of the word
"INTERMEDIATE" be arranged so that:the vowels always occupy even places?
In how many ways can the letters of the word "INTERMEDIATE" be arranged so that:
the relative order of vowels and consonants do not alter?
The letters of the word 'ZENITH' are written in all possible orders. How many words are possible if all these words are written out as in a dictionary? What is the rank of the word 'ZENITH'?
Evaluate
Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
n · n − 1Cr − 1 = (n − r + 1) nCr − 1
How many words, with or without meaning can be formed from the letters of the word 'MONDAY', assuming that no letter is repeated, if (i) 4 letters are used at a time
How many words, with or without meaning can be formed from the letters of the word 'MONDAY', assuming that no letter is repeated, if all letters are used at a time
Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.
How many words each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE?
If 35Cn +7 = 35C4n − 2 , then write the values of n.
Write the number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines.
Write the number of ways in which 12 boys may be divided into three groups of 4 boys each.
